Aristotle and Mathematics

First published Fri Mar 26, 2004

Aristotle uses mathematics and mathematical sciences in three important
ways in his treatises. Contemporary mathematics serves as a model for
his philosophy of science and provides some important techniques, e.g.,
as used in his logic. Throughout the corpus, he constructs mathematical
arguments for various theses, especially in the physical writings, but
also in the biology and ethics. Finally, Aristotle's philosophy of
mathematics provides an important alternative to platonism. In this
regard, there has been a revival of interest in recent years because of
its affinity to physicalism and fictionalisms based on physicalism.
However, his philosophy of mathematics may better be understood as a
philosophy of exact or mathematical sciences.

This article will explore the influence of mathematical sciences on
Aristotle's metaphysics and philosophy of science and will illustrate
hisuse of mathematics.

The late fifth and fourth centuries B.C.E. saw many important
developments in Greek mathematics, including the organization of basic
treatises or elements and developments in conceptions of proof, number
theory, proportion theory, sophisticated uses of constructions
(including spherical spirals and conic sections), and the application
of geometry and arithmetic in the formation of other sciences,
especially astronomy, mechanics, optics, and harmonics. The authors of
such treatises also began the process of creating effective methods of
conceiving and presenting technical work, including the use of letters
to identify parts of diagrams, the use of abstract quantities marked by
letters in proofs instead of actual numerical values, and the use of
proofs. We cannot know whether Aristotle influenced the authors of
technical treatises or merely reflects current trends.

In this context, Plato's Academy was fertile ground for controversy
concerning how we are to know mathematics (the sorts of principles, the
nature of proofs, etc.) and what the objects known must be if the
science is to be true and not vacuous. Aristotle's treatments of
mathematics reflect this diversity. Nonetheless, Aristotle's reputation
as a mathematician and philosopher of mathematical sciences has often
waxed and waned.

In fact, Aristotle's treatises display some of the technically most
difficult mathematics to be found in any philosopher before the
Greco-Roman Age. His technical failures involve conceptually difficult
areas involving infinite lines and non-homogenous magnitudes.

Commentators on Aristotle from the 2nd century on tended to
interpret Aristotle's mathematical objects as mental objects, which
made Aristotle more compatible with neo-Platonism. Later the
mechanistic movement in the late Renaissance treated Aristotle as
divorcing mathematics from physical sciences in order to drive a deeper
wedge between their views and his. Because of this, it has been very
easy to discount Aristotle as subscribing to a version of psychologism
in mathematics. These tendencies contribute to the common view that
Aristotle's views mathematics are marginal to his thought. More
recently, however, sympathetic readers have seen Aristotle as
expressing a fictionalist version of physicalism, the view that the
objects of mathematics are fictional entities grounded in physical
objects. To the extent that this view is regarded as a plausible view
about mathematics, Aristotle has regained his position.

There are two important senses in which Aristotle never presents a
philosophy of mathematics. Aristotle considers geometry and arithmetic,
the two sciences which we might say constitute ancient mathematics, as
merely the two most important mathematical sciences. His explanations
of mathematics always include optics, mathematical astronomy,
harmonics, etc. Secondly, Aristotle, so far as we know, never devoted a
treatise to philosophy of mathematics. Even Metaphysics xiii
and xiv, the two books devoted primarily to discussions of the nature
of mathematical objects, are really concerned with diffusing Platonist
positions that there are immutable and eternal substances over and
beyond sensible substances and Pythagorean positions that identify
numbers with sensible substances.

Aristotle's discussions on the best format for a deductive science in
the Posterior Analytics reflect the practice of contemporary
mathematics as taught and practiced in Plato's Academy, discussions
there about the nature of mathematical sciences, and Aristotle's own
discoveries in logic. Aristotle has two separate concerns. One evolves
from his argument that there must be first, unprovable principles for
any science, in order to avoid both circularity and infinite regresses.
The other evolves from his view that demonstrations must be
explanatory. (See subsections A, B, and C of §6, Demonstrations
and Demonstrative Sciences, of the entry
Aristotle's logic.)

An axiom (axiôma) is a
statement worthy of acceptance and is needed prior to learning
anything. Aristotle's list here includes the most general principles
such as non-contradiction and excluded middle, and principles more
specific to mathematicals, e.g., when equals taken from equals the
remainders are equal. It is not clear why Aristotle thinks one needs to
learn mathematical axioms to learn anything, unless he means that one
needs to learn them to learn anything in a mathematical subject or that
axioms are so basic that they should form the first part of one ‘s
learning.

Aristotle divides posits (thesis) into two
types, definitions and hypotheses:

A hypothesis (hupothesis) asserts one
part of a contradiction, e.g., that something is or is not.

A definition (horismos) does not assert
either part of a contradiction (or perhaps is without the assertion of
existence or non-existence).

Since a definition does not assert or deny, Aristotle probably
intends us to understand definitions as stipulations or as defining
expressions which are equivalent in some way to the defined term. The
definition of unit as ‘indivisible in quantity’ will not
presuppose that units do or do not exist. Hence, the syllogistic
premise, ‘A unit is indivisibile in quantity,’ if taken as
presupposing the existence of units will not be a definition in this
sense. Later, of course, Aristotle will allow for many other kinds of
definitions.

There are many views as to what Aristotle's hypotheses are: (i)
existence claims, (ii) any true assumption within a science, and (iii)
the stipulation of objects at the beginning of a typical proof in Greek
mathematics. Examples might be, ‘Let A be a unit,’
(where the object is stipulated to be a unit) or, more
characteristically of Greek mathematics, ‘Let there be a line
AB’ (where a line is stipulated to exist, namely
AB). In fact, all these interpretations may have a modicum of
what Aristotle means. In that case, Aristotle implies that any
assumption within a science that asserts or denies something is a
hypothesis. However, he singles out existence claims. How do existence
claims work in Aristotle's conceptions of science? From
Physics iv, we have claims such as ‘There is
place,’ and ‘There is no void.’ However, the examples
that Aristotle uses in the Posterior Analytics are claims such
as that the genus exists, or specifically that there are units, or that
there are points and lines. Aristotle also points out that sometimes
the hypothesis of the genus is omitted as too obvious. Only by
comparing these general claims with their use in Aristotelian
mathematics can we get a sense of what Aristotle means. Aristotle
intends us to understand that prior to the demonstrations in a
scientific treatise, the treatise should state starting propositions.
These include general claims broader than the science, definitions
which are stated as stipulations and not as assertions, and a claim
that the basic entities ‘exist’. What counts as an
acceptable existence claim is relative to the actual science. The
opening of a proof, ‘Let there be a line AB,’ is
an application of the basic hypothesis of the science. Since Aristotle
regards such proofs through particular lines as general proofs, the
opening claim is actually to be understood as standing for the general
claim that there are lines. This is how the hypothesis is used as a
premise. The stipulation, ‘Let there be a triangle
ABC,’ would not be a hypothesis on this interpretation,
since he holds that the existence of triangles is to be proved, so that
this instantiates a derived proposition.

A science consists of a genus (genos),
what the science is about, and a collection of attributes, what the
science says about the genus. The genus or kind is both defined and
hypothesized to exist. From his examples (points and lines for
geometry), it would seem that the genus is to be understood loosely as
the fundamental entities in the science. The attributes are defined but
are not hypothesized as existing. One must prove that the attributes
belong to various members of the genus. For example, one must prove
that triangles exist, e.g., that some [constructible] figures are
triangles.

If we take very seriously the common view that Aristotle claims that
every immediate premise of a demonstration expresses something about an
existing entity, then one may well wonder how the principles of
demonstration, axioms and posits, can be premises of demonstrations.
Existence claims and stipulations do not express something about an
existing entity. Since Aristotle calls the axioms, ‘those from
which (demonstration arises),’ some have suggested that the
axioms alone form the premises for a science and that a proof in any
science arises by placing genus terms and their definitions in the
axioms and then substituting terms like ‘triangle’ for
their definitions when they arise in proofs. However, besides pointing
to the inadequacy of the axioms for this job, it may be objected that
Aristotle also calls the principles of demonstration immediate
statements protaseis), i.e., axioms and posits. Another
possibility is that he regards even stipulations and existence claims
as premises, as well as other hypotheses, but treats the axioms as
somehow more fundamentally the source of proofs. In this case, he has a
looser conception of what counts as a premise than many readers would
expect. In any case, if his proof theory is to work at all, he must
allow many more immediate premises than one would find in the
introduction to a standard text of ancient Greek mathematics.

In the Posterior Analytics i.4, Aristotle also develops
three notions crucial to his theory of scientific claims: ‘of
every’, ‘per se’ (kath’ hauto) or ‘in
virtue of itself’ (in four ways) and ‘universally’
(katholou). Although his exposition of these notions is
tailored to his proof theory, the notions are designed also to
characterize the basic features of any scientific claim, where the
principal examples come mostly from mathematics. (See §6
Demonstrations and Demonstrative Sciences of the entry on
Aristotle's logic.)

A holds true ‘of every’ B iff
A holds of B in every case always. Note that this is
a stronger condition than is meant in the Prior Analytics by
‘A belongs to all B’. Mathematical
example: point is on every line (i.e., every line has points on it).

A is per se1 with respect to
B iff ‘A’ is in the account which gives
the essence of B. Note that Aristotle does not say that
A belongs to all B (e.g., ‘hair’ occurs
in the definition of bald, but ‘having hair’ does not
belong to a bald person), yet it is presupposed by the use Aristotle
makes of it. Aristotle allows that there are immediate statements of
the form, A belongs to no B. Mathematical examples:
‘line’ is in the definition of triangle,
‘point’ is in the definition of line.

A is per se2 with respect to
B iff ‘B’ is in the account which gives the
essence of A and A belongs to B.
Mathematical examples: straight and circular-arc belong to line, odd
and even to number. Some commentators have held that it is the
disjunction which belongs per se2 (e.g., straight or
circular-arc belongs per se2 to all line); others that the
examples are that each predicate belongs per se2 to the
subject (e.g., straight belongs per se2 to (some) line).
However, Aristotle probably knows that not all lines are straight or
circular.

A is per se3 iff
‘A’ indicates ‘a this’ (tode ti),
i.e., ‘A’ refers to just what A is. At Post. An.
i.22, Aristotle identifies the per se3 with substance, the
rock bottom of a syllogistic chain. However, one might well ask whether
there must be an analogous notion within a science. If so, A
would be per se3 if A is a basic entity in a given
science, an instance of the kind studied by the science. If so, the per
se3 items in arithmetic would be units.

A is per se4 with respect to
B iff A belongs to B on account of
A. Either no mathematical example is given or the examples are
(depending on how we read the text): straight or curved belongs to line
and odd or even belongs to number, but these may be cases of per
se2. The non-mathematical example is: in getting its throat
cut it dies in virtue of the throat-cutting.

A belongs to Buniversally iff
A belongs to all B, A belongs to B
and A belongs to B per se (in virtue of B)
and qua itself (qua B). Here the notion of ‘per
se’ seems to be slightly different from those previously
mentioned (it has been suggested that the sense is per se4),
but, in any case, is said to be equivalent to ‘qua itself’.
Perhaps we need a fifth notion of per se.

B has/is Aper se5
(i.e., in virtue of B) iff A belongs to B
qua B, i.e, there is no higher genus or kind C of
B such that A belongs to C and so to
B in virtue of belonging to C. Again, Aristotle does
not mark out per se5 as a separate notion, so that the
notion may be subsumed under per se4. Note that unlike per
se1 and per se2, per se5 is in virtue
of the subject of the predication.

The idea here seems to be that:

A belongs to Buniversally iff A belongs to all B
and A belongs per se5 to B.

An alternative (stronger?) interpretation is that:

A belongs to B universally iff A
belongs per se to all B and B belongs to all
A. In this case, A and B are called, in
modern discussions, commensurate
universals.

Aristotle describes the property that a triangle has angles equal to
two right angles as being per se5 (= per se4) and
universal, but also the property of ‘having internal angles equal
to two right angles’ as per se accidens
(kath’ hauto sembebêkôs) of triangle. It is
commonly thought that these are somehow essential
accidents. Since these follow from per se properties by
necessity, it seems strange to call them accidents at all. Sometimes,
however, it is more appropriate to think of accidents as concomitants,
the result of different demonstrative chains. Alternatively, Aristotle
frequently uses the same word to indicate consequences. In that case,
they should be called per se consequences.

It should be noted that the proof theory of Aristotle requires that
all predicates in demonstrations be either per se1 or per
se2. What is neither per se1 nor per
se2 is accidental. Hence, per se4 (or per
se5 if it is a separate notion),and per se accidens should
be reducible to these notions in any case.

Because of the formal success of his logical theory, Aristotle also
considers most mathematical proofs as having the form of a
universal affirmative syllogism, namely Barbara. (See
the section on The Syllogistic in the entry on
Aristotle's logic.)
This means
that most mathematical theorems are one thing A said of
another C and that every mathematical demonstration has a
middle term B which explains the connection between A
and C. Aristotle provides several examples of such triads of
terms in mathematics, e.g., two right angles-angles about a
point-triangle, or right angle-half two right angles-angle in a
semicircle. It has long been noted by commentators that mathematical
proofs work with a particular case through universal instantiation
(ekthesis) and then universalize to the general claim, and
that not all propositions have the form: A is said of
B, e.g., Elements 1 1, “To construct an
equilateral triangle on a given line.” A more modern objection is
that the formal theory of the syllogism as presented in Prior Analytics
1 1, 3-7 is woefully inadequate to express a theory involving
conditionals and many-many relations, as is the case with all ancient
mathematics. Nonetheless, Aristotle does think that most mathematical
proofs actually do have this form. Those that wouldn't would certainly
be negative propositions and possibly existential propositions. (We
simply do not know enough about how Aristotle conceived of the logical
form of existential propositions.) From a careful reading of the rest
of the Prior Analytics, it becomes clear that Aristotle has a
flexible notion of “one thing said of another” and that he
regards standard mathematical proofs as really being in a universal
form, which we express for purposes of comprehension as particular.

A science is defined by the genus or kind it studies and by a group of
specifiable properties which belong to that kind. Secondly, the
properties studied within a science are defined in terms of the genus
of the science (per se2). Hence, it follows that it will
commonly be impossible to prove one thing using a different science.
For one would have to prove that a property within one genus applies to
a completely different genus. Hence, every science is
autonomous. Aristotle makes this claim, however, in
the context of his rejection of Plato's view that sciences are
subordinate to knowledge of the Good. What he actually claims is more
modest. If one genus comes under another genus, it will be possible, in
some cases incumbent, to prove that a property belongs to a genus by
using a theorem from another science. In such a case the one science is
said to be under (subalternate with) the other
science.

Here are the sciences along with their relations which Aristotle
mentions in the Analytics:

Geometry

Stereometry (solid geometry)

Arithmetic

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Optics
(mathematical)

Astronomy
(mathematical)

Mechanics

Harmonics
(mathematical)

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Concerning the Rainbow

Nautical Astronomy,
Phenomena,
Empirical

Acoustical Science

Aristotle treats the science at the lowest level, descriptions of
the rainbow, astronomical phenomena, and acoustical harmonics, as
descriptive, providing the fact that something is the case, but not the
explanation, which is provided by the higher science. It is very easy
to speculate how Aristotle would fill in the relations in the table;
e.g., would he put stereometry below geometry, as Plato does in
Rep. vii? Similarly, the explanatory relation between
mathematical optics and geometry is not the same as the relation of
optics to empirical optics. This example of the rainbow seems to refer
to the argument in Meteorology iii.5, where the observed fact
that rainbows are never more than a semicircle (true in flat lands) is
explained by a proof in optics that is thoroughly geometrical in
character. Once the basic set-up and principle of reflection is
provided, the rest is geometrical.

A different situation obtains when one science is not under another
science, but some of the properties come from the other science.
Aristotle's example is the fact that that round wounds heal more slowly
[than slashes]. The medical property depends on the area of the wound
and its perimeter.

Aristotle's point about autonomy is that a theorem in arithmetic
(even less a theorem in harmonics) cannot be used to prove something in
geometry. Here, arithmetic is probably understood as the number theory
found in Euclid, Elements vii-ix, and not mere calculation of
numbers, which, of course, is used in geometry. This allows the
anti-Platonic point that theorems about the beautiful and theorems in
mathematics have nothing to do with one another, even if some theorems
are beautiful.

Elsewhere (esp. Physics ii.2 and Metaphysics
xiii.3),Aristotle provides different accounts of the relations between
mathematical sciences.

Aristotle's principal concern in discussing ontological issues in
mathematics is to avoid various versions of platonism. Aristotle shares
with Plato the view that there are objects of understanding, that these
must be universal and not particular and that they have to satisfy
certain “Parmenidean” conditions, such as being unchanging
and eternal. However, Aristotle rejects the view of Plato that objects
of understanding are separate from particulars. This is a general
problem in Aristotle's metaphysics. However, in the case of
mathematical objects, there are three important difficulties. First, if
physical objects are the objects of mathematical understanding and
satisfy the standard definitions of line, circle, etc., then they
manifestly fail in two ways (cf. Met. iii 2 997b25-8a19):

The physical straight lines we draw are not straight; a physical
tangent line does not really touch a circle at a point. In other words,
physical objects fail to have the mathematical properties we study.
This is the problem of precision.

Physical mathematical objects lack properties which we require of
objects of understanding. They are not separate or independent of
matter. Hence, they are not eternal or unchanging. This is the problem
of separability.

Although these two problems are distinct, Aristotle may hold that
this failure is at least partly responsible for the failure of
mathematical objects to have the mathematical properties we study.
Platonic Forms fail in a third way.

Suppose that there is a Form for each kind of triangle. There still
would be only one Form for each kind. A mathematical theorem about
diagonals of rectangles might mention two equal and similar triangles
which are, nonetheless, distinct. Mathematical sciences require many
objects of the same sort. This is the problem of
plurality (cf. Met. iii 2 and Met.
iii.1-2).

A fourth problem is not explicitly stated by Aristotle, but is
clearly a presupposition of his discussion.

An account of mathematics should not impinge on mathematical
practice so as to make it incoherent or impossible. If mathematicians
talk about triangles, numbers, etc., the account of mathematical
objects should at least explain the discourse. This is the problem of
non-revisionism (sometimes also called
naturalism). So Aristotle says (Met. xiii.3
1077b31-33) of his own account of mathematics that “it is
unqualifiedly true to say of the mathematicals that thay exist and are
such as they <the mathematicians say>”
(cf. Phys. iii.7 207b27-34 for an application of the
principle).

To solve the problems of separation and precision, contemporary
philosophers such as Speusippus and possibly Plato posited a universe
of mathematical entities which are perfect instances of mathematical
properties, adequately multiple for any theorem we wish to prove, and
separate from the physical or perceptible world. Aristotle calls them
mathematicals or intermediates,
because they are intermediate between the Forms and physical objects,
in as much as they are perfect, eternal, and unchanging like the Forms,
but multiple like physical objects (cf., for example, Met. i.6
987b14-18, iii 2, xiii.1-2). This solution is the ancestor of many
versions of platonism in mathematics.

Aristotle's rejection of intermediates involves showing that their
advocates are committed to an unwieldy multiplicity of mathematical
universes, at least one corresponding to each mathematical science,
whether kinematics, astronomy, or geometry. However, he also sets out
to show that such ontologies are not merely pleonastic, but also that
an alternative account can be given free of all the difficulties
mentioned. In other words, Aristotle's strategy is best seen as
diffusing some versions of platonism.

Aristotle rejects a compromise as merely compounding this
difficulty, the view that either Forms or intermediates are immanent in
things (separate but coextensive), since these different worlds will
now have to exist bundled together.

Aristotle occasionally refers to mathematical objects as things by,
in, from, or through removal (in different works Aristotle uses
different expressions: ta aphairesei, ta en
aphairesei, ta ex aphaireseôs, ta di’
aphaireseôs). It is also clear that this usage relates to
logical discussions in the Topics of definitions where one can
speak of adding a term or deleting a term from an expression and seeing
what one gets as a result. Our principal task is to explain what this
logical/psychological removal is and how it solves the four puzzles.
Aristotle starts with the class of perceptible or physical
magnitudes. The examination of these is a part of physics (cf.
Physics iii.4). The ontological status of these does not
concern him, but we may suppose that they consitute the category of
quantities: the bodies, surfaces, edges, corners, places, and times,
sounds, etc. of physical substances (Categories 6).

In the Analytics, where the notion of matter is absent,
Aristotle begins with a particular geometrical perceptible figure. What
is removed is its particularity and all that comes with this, including
its being perceptible. What is left then is a universal of some sort.
Aristotle also does not seem to think in this work that there is any
conflict between the plurality problem and thinking of all terms in a
mathematical deduction as universals. However, since he allows that a
term can be a very complex expression, it can designate a rich complex
for which there would probably be no corresponding Form in a Platonic
theory.

Elsewhere, Aristotle usually seems to mean that the attributes not a
part of the science are removed. What is left may be particular, a
quasi-fictional entity. It is the status of this entity which leads to
much controversy? Is it a representation in the soul or is it the
perceptible object treated in a special way? Ancient and medieval
readers tended to take the former approach in their interpretation of
Aristotle, that the object left is a stripped down representation with
only the required properties. (Cf. Mueller (1990) and neo-Platonic
foundations for these interpretations of mathematics, such as Proclus
in his commentary on Euclid.)

Most modern readers, perhaps influenced by the critiques of Berkeley
and Hume against the first and certainly less committed to
neo-Platonism, take the second approach. The objects studied by
mathematical sciences are perceptible objects treated in a special way,
as a perceived representation, whether as a diagram in the sand or an
image in the imagination. Furthermore, perhaps as a response to Frege's
devastating critique of psychologism and Husserl's first attempt at a
psychological account of arithmetic, some have suggested that Aristotle
has no need for a special faculty of abstracting. Rather the mind is
able to consider the perceived object without some of its properties,
such as being perceived, being made of sand, etc. However, this is
analogous to the logical manipulation of definitions, by considering
terms with or without certain additions. Hence, Aristotle will
sometimes call the material object, the mathematical object by adding
on. As a convenience, the mind conceives of this as if the object were
just that. On this view, abstraction is no more and no less
psychological than inference.

Conceptually, we might think of the process as the mind rearranging
the ontological structure of the object. As a substantial artifact,
what-it-is, the sand box has certain properties essentially. The figure
drawn may be incidental to what-it-is, i.e., an accident. In treating
the object as the figure drawn, being made of sand is incidental to it.
Hence, ‘things by removal’ may be one way of explaining
perceptible magnitudes qua lengths. This is the concept which
does most of the work for Aristotle.

In his discussions of precision, Aristotle states that those
sciences which have more properties removed are more precise.
Arithemetic, about units, is more precise than geometry, since a point
is a unit having position. A science of kinematics (geometry of moving
magnitudes) where all motion is uniform motions is more precise than a
science that includes non-uniform motions in addition, and a science of
non-moving magnitudes (geometry) is more precise than one with moving
magnitudes. However, one might infer that ‘precision’ here
means nothing more than ‘clarity’ (or perhaps
‘refinement’, with all its ambiguity). Does this concept of
‘precision’ provide a framework for solving the problem of
precision?

Aristotle solves the separability problem with a kind of
fictionalism. The language and practice of mathematicians is legitimate
because we are able to conceive of perceptible magnitudes in ways that
they are not. The only basic realities for Aristotle remain substances,
however we are to conceive them. A primary characteristic of substances
is that they are separate. Yet we are able to speak of a triangle, a
finite surface, merely the limit of a body, and hence not separate, as
if it were separate (hôs kekhôrismenon). It is a
subject in our science (in our discourse in the science). The mental
and logical mechanism by which we accomplish this is the core of
Aristotle's strategy in diffusing platonisms.

The word Aristotle uses is commonly translated with the English word
‘qua’ which itself translates the Latin relative
pronoun ‘qua’, but with one important grammatical
difference. The English adverb is normally followed by a noun
phrase.

As a relative adverbial pronoun in the dative case, the Greek word
captures all possible meanings of the dative, including,
‘where’, ‘in the manner that’,
‘by-means-of-the-fact-that’, or
‘in-the-respect-that’. Some have suggested translating it
with the word ‘because’, although it is arguable that the
English word at best intersects with the appropriate Greek meaning
(perhaps ‘just or precisely because’ works better. Hence,
‘XquaY’ should be understood
as elliptic for:

‘X in the respect that X is
Y’

or

‘X by means of the fact that X is
Y’ (or ‘X precisely because X is
Y’).

In the context of a scientific claim, ‘X
by-means-of-the-fact-that or in-the-respect-that X is
Y is F’ maintains that ‘Y is
F’ is a theorem, where Y is the most universal
or appropriate subject for F, and that X is
F in virtue of the fact that X is Y. For
example, Figure ABC qua triangle has internal angles equal to
two right angles, but qua right triangle has sides
AB2 + BC2 = AB2.

In the case where we examine or study an object XquaY or X in-the-respect-that X is
Y, we study the consequences that follow from something's
being an Y. In other words, Y determines the logical
space of what we study. If X is a bronze triangle (a
perceptible magnitude), to study Xqua bronze will be
to examine bronze and the properties that accrue to something that is
bronze. To study Xqua triangle is to study the
properties that accrue to a triangle. Unless it follows from
something's being a triangle that it must be bronze, the property of
being bronze will not appear in one's examination.

Note that there is no necessity that ‘qua’
operators be of the form ‘quaY’, where
Y is a noun phrase. For example, Aristotle says (De
anima iii.4.429b25-6) that two things affect and are affected
“qua something in common belongs to both.”
Similarly, as evidence that ‘qua’ does not in
these contexts always mean ‘because’ (usually, the context
is too ambiguous to precisely decide whether it means
‘because’ or ‘in the respect that’), consider
Nicomachean Ethics i.3.1102b8-9, “Sleep is an inactivity
of the soul qua it is called good or bad,” but certainly
not because it is.

With only one or two possible exceptions, it seems that whenever
Aristotle speaks of F(X) quaG(X), G(X) must be true. We can
study a perceptible triangle qua triangle because it is a
triangle. For convenience, we can call this principle
qua-realism.

The Account of Mathematical Objects with
‘Qua’. We begin with perceptible
magnitudes. These are volumes, surfaces, edges, and corners. They
change in position and size. They are made of some material and are the
quantities of substances and their interactions. The volumes, surfaces,
and edges have shape. Times and corners do not. Different sciences
treat different perceptible magnitudes qua different
things.

Moreover, since there are many perceptible magnitudes, there will be
enough, qua line, to prove any theorem that involves lines.
The plurality problem is trivially solved.

The separability problem is solved because if we examine XquaY, we will talk about Y as if it is a
separate entity, as a subject, and will pay no attention to the way in
which we captured Y through a description
‘X’, in the sense that only the residue of the
qua-filter are studied. The science will speak of Y.
This too will not interfere with mathematical practice and so will not
violate non-revisionism.

In Metaphysics vi.1, Aristotle argues that physics concerns
things which have change, but are substances, that at least some of the
things that mathematics is about do not change and are eternal but are
not substances (exceptions would probably include stars and spheres in
mathematical astronomy and bodies in mathematical kinematics), while
first philosophy or theology is about things which are substances but
do not change and are eternal. We can now characterize the way in which
mathematical objects are eternal and lack change. Namely, generation
and change are not among the predicates studied by geometry or
arithmetic. Hence, it is correct to say that qua lines,
perceptible lines lack generation, destruction, and change (with
appropriate provisos for kinematics and mathematical astronomy).

Whether the precision problem is also solved and how it is solved is
more controversial. On the ancient and medieval interpretation, the
problem of precision is solved by allowing mental representations to be
as precise as one chooses. The contemporary interpretation of
considering Aristotle's mathematical objects as physical object treated
in a special way has a more difficult task. There are five ways in
which Aristotle may attempt to solve the precision problem.

Many scholars today seem to hold to a view that for Aristotle if
one can speak of XquaY, then X
must be Y precisely. This means that any theorem about
triangles will only hold of the rare perfect triangles, wherever they
may be (a thesis once suggested by Descartes).

To increase the number of instances of exact triangles in the
ontology, some scholars turn to Met. xiii.3, where Aristotle
says that being is said in two ways, the one in actualilty and one
materially, he may be pointing to the fact that mathematical entities
exist in continua as potentialities. Hence, a perfect line exists
potentially in the sand, even if the one I have drawn is not. (Some
have also seen support for this in Met. ix.9.) Hence, although
there may be no actual triangles right now, at least there are an
infinity of potential ones. The difficulty is that the argument is not
about precision. It concerns an objection to Aristotle that man
qua man is indivisible, but geometry studies man qua
divisible. Since man is not divisible, the principle of
qua-realism, that if one can study XquaY then X is Y, is violated. Aristotle says
that man is in actuality indivisible (you cannot slice a man in two and
still have man or men), but is materially divisible. It is enough that
X is Y materially or in actuality to study XquaY. Nonetheless, the solution to the puzzle could
point to an Aristotelian solution to the problem of precision.

Alternatively, it is arguable that Aristotle allows that
‘X is Y’ may be true only imprecisely.
For example, I may study a triangle in a diagram ABCqua triangle, but ABC is only a triangle imprecisely.
Many Hellenistic treatises involving applied sciences set up convenient
but false premises for the purposes of mathematical manipulation,
including, notably, Aristotle's own account of the rainbow (Meteorology
iii.5). Hence, an appeal to potentialities to get more exact triangles
will do nothing to eliminate these apparent violations of
qua-realism.

In providing his hierarchy of precision in sciences, Aristotle may
think that from filtering out more properties one gets greater
precision. One finds more precise straight lines in geometry than in
kinematics. Besides the obscurity of the position, it is not clear that
he intends any such thing (see Section 7.2 above).

One possibility is that Aristotle thinks that if a description has
more properties removed from consideration, the entity studied is more
precise in that there will be instances materially or actually that
exactly exhibit satisfy qua-realism. For example, there are
precise instances of units or of corners or points so that arithmetic
and geometry are precise, while astronomy might not be so precise,
since the planets are imprecisely points, but are studied qua
points.

Our difficulty is that while Aristotle raises the problem of
precision, he does not explicitly explain his solution to it.

Perceptible magnitudes have perceptible matter. A bronze sphere is a
perceptible magnitude. For solving the plurality problem, Aristotle
needs to have many triangles with the same form. Since perceptible
matter is not part of the object considered (in abstraction or
removal), he needs to have a notion of matter which is the matter of
the object: bronze sphere MINUS bronze (perceptible matter). Since this
object must be a composite individual to distinguish it from other
individuals with the same form, it will have matter. He calls such
matter intelligible or mathematical matter. Aristotle has at least four
different conceptions of intelligible matter in the middle books of the
Metaphysics, Physics iv, and De anima i:

The form of a magnitude is its limit (Metaphysics v.17);
hence, the matter is what is between the limits of the magnitude, its
extension (Physics iv 2).

Matter is the genus, e.g., in the sense that magnitude (and not
perceptible-magnitude) is the kind for triangles (e.g.,
Metaphysics v.28, viii.6).

The ‘non-perceptible’ matter of a perceptible
magnitude, which is in the perceptible matter (Metaphysics
vii.10, cf. De anima i.1).

The parts of a mathematical object which do not occur in the
definition of the object, e.g., acute angle is not in the definition of
right angle, but is a part of it and so is a non-perceptible material
part of the angle (Metaphysics vii.10, 11).

(1) and (3) are compatible; (2) may be a separate notion having more
to do with the unity of definition and seems incompatible with (4);
Aristotle treats (3) and (4) as the same notion. Since Aristotle's
concern in discussing (4) is with the nature of the parts of
definitions and not with questions of extended matter, it is unclear
whether the non-definitional parts are potential extended parts or
merely forms of extended parts, although the former seems more
plausible.

Ancillary to his discussions of being qua being and theology
(Metaphysics vi.1, xi.7), Aristotle suggests an analogy with
mathematics. If the analogy is that there is a super-science of
mathematics coverying all continuous magnitudes and discrete
quantities, such as numbers, then we should expect that Greek
mathematicians conceived of a general mathematical subject as a
precursor of algebra, Descartes' mathesis universalis (universal
learning/mathematics), and mathematical logic.

Aristotle reports (Posterior Analytics. i.5, cf.
Metaphysics xiii.2) that whereas mathematicians proved
theorems such as a : b = c : d
=> a : c = b : d (alternando)
separately for number, lines, planes, and solids, now there is one
single general or universal proof for all (see Section 3). The
discovery of universal proofs is usually associated with Eudoxus'
theory of proportion. For Aristotle this creates a problem since a
science concerns a genus or kind, but also there seems to be no kind
comprising number and magnitude. Some scholars have proposed that a
universal science of ‘posology’ (a science of quantity)
takes the whole category of quantity as its subject.

Aristotle seems more reticent, describing the proofs as concerning
lines, etc., qua having such and such increment (An. Post.
ii.17). He seems to identify such a super-mathematics
(Metaphysics vi.1, xi.7), but seems to imply that it does not
take a determinate kind as its subject. Another possibility is that the
common science has theorems which apply by analogy to the different
mathematical kinds.

Elsewhere (Metaphysics xi.4), where Aristotle builds an
analogy with the science of being qua being, he seems to suggest that
universal proofs of quantities (here too including numbers) concern
continuous quantity (unlike the similar passage in Metaphysics
iv.3). If so and if this is by Aristotle, it would correspond to the
general theory of proportions as it comes down to us. One may well
wonder if scholars have been led astray by a hyperbole about universal
proofs.

Ironically, extant Greek mathematics shows no traces of an
Aristotelian universal mathematics. The theory of ratio for magnitudes
in Euclid, Elements v is completely separate from the
treatment of ratio for number in Elements vii and parts of
viii, none of which appeals to v, even though almost all of the proofs
of v could apply straightforwardly to numbers. For example, Euclid
provides separate definitions of proportion (v def. 5, and vii def.
20). Compare the rule above (alternando),which is proved at v.16, while
the rule follows trivially for numbers from the commutivity of
multiplication and vii.19: ad = bc ⇔ a
: b = c : d.

In Plato's Academy, some philosophers suggested that lines are composed
of indivisible magnitude, whether a finite number (a line of
indivisible lines) or a infinite number (a line of infinite points).
Aristotle builds a theory of continuity and infinite divisibility of
geometrical objects. Aristotle denies both conceptions. Yet, he needs
to give an account of continuous magnitudes that is also free from
paradoxes that these theories attempted to avoid. The elements of his
account may be found principally in Physics iv.1-5 and v.1 and
vi. Aristotle's account pertains to perceptible magnitudes. However, it
is clear that he understands this to apply to magnitudes in mathematics
as well.

Aristotle has many objections to thinking of a line as composed of
actual points (likewise, a plane of lines, etc.), including:

No point in a line is adjacent to another point.

If a line is composed of actual points, than to move a distance an
object would have to complete an infinite number of tasks (as suggested
by Zeno's arguments against motion)

To say that a line is comprised of an infinity of potential points
is no more than to say that a line may be divided (with a line-cutter,
with the mind, etc.) anywhere on it, that any potential point may be
brought to actuality. The continuity of a line consists in the fact
that any actualized point within the line will hold together the line
segments on each side. Otherwise, it makes no sense to speak of a
potential point actually holding two potential lines together.

Suppose I have a line AB and cut it at C. The
lines AC and CB are distinct. Is C one point
or two?

C is one point in number.
C is two points in its being or formula
(logos).

This merely means that we can treat it once or twice or as many times
as we choose. Note that Aristotle says the same thing about a
continuous proportion. in a : b = b :
c, b is one magnitude in being, but is used as two.

Greek mathematicians tend to conceive of number (arithmos) as
a plurality of units. Perhaps a better translation, without our deeply
entrenched notions, would be ‘count’. Their conception
involves:

A number is constructed out of some countable entity, unit
(monas).

Numbers are more like concatenations of units and are not sets. To
draw a contrast with modern treatments of numbers, a Greek pair or a
two is neither a subset of a triple, nor a member of a triple. It is a
part of three. If I say that ten cows are hungry, then I am not saying
that a set is hungry. Or to point to another use of ‘set’, my 12 piece
teaset is in a cabinet, not in an abstract universe. So too, these ten
units are a part of these twenty units:

One (a unit) typically is not a number (but Aristotle is ambivalent
on this), since a number is a plurality of units.

At least in theoretical discussions of numbers, a fractional part
is not a number.

In other words, numbers are members of the series: 2, 3, …,
with 1 conceived as the ‘beginning’ (archê)
of number or as the least number.

In early Greek mathematics (5th century), numbers were represented
by arrangements of pebbles. Later (at least by the 3rd century BCE)
they were represented by evenly divided lines.

For Aristotle and his contemporaries there are several fundamental
problems in understanding number and arithmetic:

The precision problem of mathematicals is similar in the case of
geometrical entities and units (see Section 6). Consider, for example,
Plato's discussion of incompatible features of a finger as presenting
one or two things to sight. Aristotle deals with the problem in his
discussion of measure (see Section 10.1).

The separability problem is the same as for geometry (see Section
6).

The plurality problem of mathematicals (Section 6) is similar in
the case of geometrical entities and units, with some differences. To
count ‘perceptible’ units, or rather units from abstraction
(cf. Section 7.1), one needs some principle of individuating units,
what one is counting, whether cows or categories of predication.
Aristotle says that one can always find an appropriate classification
(we may assume that some classifications would be fairly convoluted,
but that this is at best an aesthetic and not a logical problem). For
units, one will use the same principle that allows one to individuate
triangles. This is why Aristotle can describe a point as
unit-having-position. Arithmetic involves the study of entities qua
indivisible.

The unity problem of numbers: This problem
bedevils philosophy of mathematics from Plato to Husserl. What makes a
collection of units a unity which we identify as a number? It cannot be
physical juxtaposition of units. Is it merely mental stipulation?

Aristotle does not seem bothered by:

The overlap problem: What guarantees that when I
add this 3 and this 5 that the correct result is not 5, 6, or 7, namely
that some units in this 3 are not also in this 5.

Aristotle presents three Academic solutions to these problems. Units
are comparable if they can be counted together (such as the ten cows in
the field). Units are not comparable, if it is conceptually impossible
to count them together (a less intuitive notion).

Incomparable Units: Form numbers are conceived as
ordinals, with units conceived as being well ordered. What makes this
number 3 is not that it is a concatenation of three units, but that its
unit is the third unit in this series of units. Hence, it is simply
false that there is a unity of the first three units forming a number
three. What makes an ordinary concatenation, e.g., a herd of cows, ten
cows is that they can be counted according to the series of
Form-numbers. The notion of incomparable numbers lacks the basic
conception of numbers as concatenations of units.

Comparable/Incomparable Units: Form numbers are,
of course, special. Each is a complete unity of units. For example, the
Form of 3 is a unity of three units. Since it is a unity, it cannot be
an accident that these three units form this unity. They are comparable
with each other in the sense that together they comprise Three Itself
and perhaps cannot be conceived separately. Hence, they cannot be parts
of any other Form number. We cannot take 2 units from the Three Itself
and add them to 4 units in the Six itself, to get a Form-number of the
Seven itself. Myles Burnyeat once suggested an analogy with a sequence
of playing cards of one suit, say
diamonds.[1]
Each card from ace (unit) to ten contains the appropriate diamonds
(from 1 to 10) on each card, unified by their being on their
particular card. Yet we don't count up two diamonds from the deuce and
two from the trey, but treat each card as a complete unity.

Comparable Units: These are intermediate or
mathematical numbers (see Section 6). There is a unlimited number of
units (enough to do arithmetic), which are arranged and so forth.
Comparable numbers solve the plurality problem, but not the unity
problem.

Aristotle reports that some Academics opted for a version of
Incomparable or Comparable/Incomparable Units to solve the unity
problem and introduced comparable units as the objects of mathematical
theorems, e.g., given some comparable units, they are even if they can
be divided in half, into two concatenations corresponding to
(participating in) the same Form-number.

10.2.1 Background

Greeks used an Egyptian system of fractions. With the exception of 2/3,
all fractions are proper parts which modern readers will see as unit
fractions: 1/n. For example, 2/5 is 1/3 1/15 (that is, the sum of 1/3
and 1/15). Additionally, Greeks used systems of measure, as we do, with
units of measure being divided up into more refined units of measure. 1
foot is 16 finger-widths (inches). Hence, one can always eliminate
fractions by going, as we do, to a more refined measure (1, 1/2, 1/4
feet or 1 foot 12 inches). This feature of measure may be reflected in
Plato's observation (Rep. vii) that in arithmetic, one can
always eliminate parts of units.

10.2.2 Aristotle's views

Since Aristotle (esp. Metaphysics x.1-2) treats measurement under his
discussions of units (and hence number), it turns out that the
precision problem becomes a problem of sorting out precise units of
measure. Hence, in the case discrete quantities, such as cows, the unit
is very precise, one cow. In the case of continuous quantities, the
most precise unit of time is the time it takes for the fixed stars (the
fastest things in the universe) to move the smallest perceptible
distance. But a point or indivisible with position removed or a cow qua
unit, i.e., a mathematical unit, is precise.

Aristotle's treatment of time (Physics iv.10-14) includes some
observations about numbers which come closest to being an acount of
number. Aristotle defines time as the number or count of change and
then proceeds to distinguish two senses of number, what is counted
(e.g., ten cows as measured by the cow-unit, or ten feet as measured by
the foot-unit) and that by which we count. Time is number in the first
sense, not as so-many changes, but as so-much change as measured by a
unit of change. Aristotle clarifies the distinction between what is
counted (or is countable) and that by which we count. These five black
cats (number as what is counted) are different from these five brown
cats, but their number (that by which we count) is the same.

But what are the numbers by which we count? Aristotle says nothing, but
we may speculate that the five by which we count is the single formal
explanation of what makes the five black cats five and what makes the
five brown cats five. Hence, Aristotle probably subscribes to an
Aristotelian version of the distinction between intermediate and
Form-numbers.

Aristotle's discussion of time also gives us some insight into the
unity problem. What gives the five black cats unity is just that they
can be treated as a unity. From this it follows for Aristotle that
there can be no number without mind. Nothing iscountable unless there
exists a counter.

It is often supposed, for Aristotle, mathematical explanation plays no
role in the study of nature, especially in biology. This conception is,
most of all, a product of anti-scholastics of the late Renaissance, who
sought to draw the greatest chasm between their own mechanism and
scholasticism. Mathematics plays a vital role in both. The principal
way in which mathematics enters into biological explanation is through
hypothetical necessity:

If X is to have feature Y (which is good
for X), then it is a feature of its matter that Z be
the case.

Z may be a constraint determined by a mathematical fact.
For example, animals by nature do not have an odd number of feet. For
if one had an odd number of feet, it would walk awkardly or the feet
would have to be of different lengths (De incessuanimalium 9).
To see this, imagine an isosceles triangle with a altitude drawn.

Aristotle famously rejects the infinite in mathematics and in
physics, with some notable exceptions. He defines it thus:

The infinite is that for which it is always possible to
take something outside.

Implicit in this notion is an unending series of magnitudes, which will
be achieved either by dividing a magnitude (the infinite by division)
or by adding a magnitude to it (the infinite by addition). This is why
he conceives of the infinite as pertaining to material explanation, as
it is indeterminate and involves potential cutting or joining (cf.
Section 7.5).

Aristotle argues that in the case of magnitudes, an infinitely large
magnitude and an infinitely small magnitude cannot exist. In fact, he
thinks that universe is finite in size. He also agrees with Anaxagoras,
that given any magnitude, it is possible to take a smaller. Hence, he
allows that there are infinite magnitudes in a different sense. Since
it is always possible to divide a magnitude, the series of division is
unending and so is infinite. This is a potential, but never actual
infinite. For each division potentially exists. Similarly, since it is
always possible to add to a finite magnitude that is smaller than the
whole universe continually smaller magnitudes, there is a potential
infinite in addition. That series too need never end. For example, if I
add to some magnitude a foot board, and then less than a half a foot,
and then less than a fourth, and so forth, the total amount added will never
exceed two feet. Aristotle claims that the mathematician never needs
any other notion of the infinite.

However, since Aristotle believes that the universe has no beginning
and is eternal, it follows that in the past there have been an infinite
number of days. Hence, his rejection of the actual infinite in the case
of magnitude does not seem to extend to the concept of time.

As philosophers usually do, Aristotle cites simple or familiar examples
from contemporary mathematics, although we should keep in mind that
even basic geometry such as we find in Euclid's Elements would
have been advanced studies. The average education in mathematics would
have been basic arithmetical operations (possibly called
logistikê) and metrological geometry (given certain
dimensions of a figure, to find other dimensions), such as were also
taught in Egypt. Aristotle does allude to this sort of mathematics on
occasion, but most of his examples come from the sort of mathematics
which we have come to associate with Greece, the constructing of
figures from given figures and rules, and the proving that figures have
certain properties, and the ‘discovery’ of numbers with
certain properties or proving that certain classes of numbers have
certain properties. If we attend carefully to his examples, we can even
see an emerging picture of elementary geometry as taught in the
Academy. In the supplement are provided twenty-five of his favorite
propositions (the list is not exhaustive).

Aristotle also makes some mathematical claims that are genuinely
problematic. Was he ignorant of contemporary work? Why does he ignore
some of the great problems of his time? Is there any reason why
Aristotle should be expected, for example to refer to conic sections?
Nonetheless, Aristotle does engage in some original and difficult
mathematics. Certainly, in this Aristotle was more an active
mathematician than his mentor, Plato.